1. If there are any necessary, eternal, omnipresent, immutable and immaterial objects, then this fact is most parsimonious with theism. (Premise)
2. Laws of logic exist. (Premise)
3. The laws of logic are necessary, eternal, omnipresent, immutable and immaterial. (Premise)
4. Therefore, the most parsimonious hypothesis with the laws of logic is theism. (From 1 - 3)
A rejection of (2) would result in a form of anti-realism. Due to the indispensability of the laws of logic, I think we're safe in asserting the truth of (2). (3) seems obviously true just based on an analysis of what the laws of logic are. They cannot be falsified at any time or any place, and so they are necessary, eternal and omnipresent. Moreover, the laws of logic cannot change. It's not as if the law of non-contradiction ("something cannot both be and not be at the same time and in the same sense") can suddenly become the opposite of what it is. For, that would actually presuppose the law of non-contradiction. Finally, the laws of logic aren't things we can just reach out and touch. They are immaterial, e.g. not extended in space.
The atheist's best bet is to somehow undermine (1). It's at this point that Augustine adopts conceptualism in order to refute the atheist's final option: Platonism.